拟阵的多项式不变量

Matroid is a combinatorial structure that axiomize the notions of linear independence of vector spaces and cycle structure of graphs. In the past decade, three polynomial invariants associated to matroid have been well studied. Huh etc. applied Hodge theory to characteristic polynomials of matroid and gave a positive answer to Rota's conjecture; Moci etc. established theoretical foundations of arithmetic matroids, matroids over rings, and their characteristic polynomials and Tutte polynomials; Analogous to the classical representation theory, Proudfoot etc. introduced Kazhdan-Lusztig polynomials of matroid. Motivated by their works, our purpose of this proposal is to investigate more on characteristic polynomials, Tutte polynomials, and Kazhdan-Lusztig polynomials of matroids, arithmetic matroids and matroids over rings.

拟阵(Matroid)以公理化形式推广了向量组的线性无关结构和图中的圈结构。近十年来，拟阵多项式理论发展迅速，Huh等人将代数几何中的Hodge理论引入到拟阵特征多项式的研究中并解决了Rota猜想, Moci等人提出了算术拟阵和环上拟阵的公理化定义及其特征多项式和Tutte多项式,Proudfoot等人在拟阵上模拟表示论中的经典理论定义了拟阵的Kazhdan-Lusztig多项式。本项目将在这三组团队的工作基础上，结合申请人和项目组成员在这方面已有的积累，对拟阵的特征多项式，Tutte多项式，以及Kazhdan-Lusztig多项式做更深入的研究。

近年来，国际上形成了多个研究团队关注拟阵的组合分类以及相关不变量的研究。 本项目主要研究了可表示拟阵以及超平面配置的组合分类及一系列组合不变量，包括其交半格、特征多项式、Tutte多项式、Whitney多项式，两类Whitney数等。同时，还研究了这些组合不变量对于拟阵或超平面配置的单元扩张、单元对偶扩张、收缩、k-维子空间限制等分类问题中的变化规律。共形成了10篇科研论文，发表于组合数学著名期刊, 包括J. Combin. Theory Ser. A， Adv. in Appl. Math.，Math. Proc. Cambridge Philos. Soc.，Proc. Amer. Math. Soc.，Discrete Math.，Electron. J. Combin.等等。